A statistical treatment of the gamma-ray burst "no host galaxy" problem. I. Methodology

If gamma-ray bursts originate in galaxies at cosmological distances, the host galaxy should be detected if a burst error box is searched deeply enough; are the host galaxies present? We present and implement a statistical methodology that evaluates whether the observed galaxy detections in a burst's error box are consistent With the presence of the host galaxy, or whether all the detections can be attributed to unrelated background galaxies. This methodology requires the model-dependent distribution of host-galaxy fluxes. While our methodology was derived for galaxies in burst error boxes, it can be applied to other candidate host objects (e.g., active galaxies) and to other types of error boxes. As examples, we apply this methodology to two published studies of burst error boxes. We find that the nine error boxes observed by Larson and McLean are too large to discriminate between the presence or absence of host galaxies, while the absence of bright galaxies in the four significantly smaller error boxes observed by the Hubble Space Telescope (Schaefer et al.) does confirm that there is a "no host galaxy" problem within the "minimal" host-galaxy model.